We discuss a system of strengthenings of "$\aleph_\omega$ is J\'onsson" indexed by real numbers, and identify a strongest one. We give a proof of a theorem of Silver and show that there is a barrier to weakening its hypothesis.
Axioms for the Real Numbers: A Constructive Approach
A question of van den Dries and a theorem of Lipshitz and Robinson: Not everything is standard
Connectedness in structures on the real numbers: o-minimality and undecidability
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